Multi-Element Stochastic Galerkin Method Based on Edge Detection for Uncertainty Quantification of Discontinuous Responses

2020 ◽  
Vol 5 (4) ◽  
Author(s):  
Shigetaka Kawai ◽  
Akira Oyama
Keyword(s):  
Edge Detection ◽  
Gibbs Phenomenon ◽  
Test Problems ◽  
Time Step ◽  
Computational Costs ◽  
Generalized Polynomial ◽  
Minimal Decomposition ◽  
Resolution Level ◽  
Stochastic Galerkin ◽  
Discontinuous Responses

Abstract We propose a new multi-element generalized polynomial chaos (MEgPC) method to minimize the computational costs required for the existing MEgPC to circumvent the Gibbs phenomenon in the presence of discontinuities in a random space. The proposed method uses edge detection to capture the discontinuous behavior of a solution with minimal decomposition. In contrast, the existing MEgPC iterates splitting the random space into two equal parts until achieving a sufficient resolution level. We take advantage of the fact that the stochastic Galerkin (SG) methods facilitate adaptive refinement of the decomposition at every time-step during a computation for the proposed method. The numerical experiments for two-test problems demonstrate the performance of the proposed method. The results show that the proposed method is consistently more accurate than conventional methods for sufficiently high polynomial orders with minimal additional computational costs to capture discontinuities.

scholarly journals THE NUMERICAL TREATMENT OF NONLINEAR FRACTAL–FRACTIONAL 2D EMDEN–FOWLER EQUATION UTILIZING 2D CHELYSHKOV POLYNOMIALS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040042
Author(s):  
M. HOSSEININIA ◽  
M. H. HEYDARI ◽  
Z. AVAZZADEH
Keyword(s):  
Approximate Solution ◽  
Finite Difference ◽  
Test Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Time Step ◽  
Discrete Method ◽  
Difference Formula ◽  
Fowler Equation ◽  
Chelyshkov Polynomials

This paper develops an effective semi-discrete method based on the 2D Chelyshkov polynomials (CPs) to provide an approximate solution of the fractal–fractional nonlinear Emden–Fowler equation. In this model, the fractal–fractional derivative in the concept of Atangana–Riemann–Liouville is considered. The proposed algorithm first discretizes the fractal–fractional differentiation by using the finite difference formula in the time direction. Then, it simplifies the original equation to the recurrent equations by expanding the unknown solution in terms of the 2D CPs and using the [Formula: see text]-weighted finite difference scheme. The differentiation operational matrices and the collocation method play an important role to obtaining a linear system of algebraic equations. Last, solving the obtained system provides an approximate solution in each time step. The validity of the formulated method is investigated through a sufficient number of test problems.

Numerical simulation of higher-order diffusion-wave equations of variable coefficients using the meshless spectral method

2020 ◽  
pp. 2150010
Author(s):  
Manzoor Hussain ◽  
Sirajul Haq
Keyword(s):  
Wave Equations ◽  
Interpolation Method ◽  
Variable Coefficients ◽  
Higher Order ◽  
Implicit Time ◽  
Test Problems ◽  
Time Step ◽  
Diffusion Wave ◽  
Time Step Size ◽  
Time Stepping

In this paper, meshless spectral interpolation technique using implicit time stepping scheme is proposed for the numerical simulations of time-fractional higher-order diffusion wave equations (TFHODWEs) of variable coefficients. Meshless shape functions, obtained from radial basis functions (RBFs) and point interpolation method (PIM), are used for spatial approximation. Central differences coupled with quadrature rule of [Formula: see text] are employed for fractional temporal approximation. For advancement of solution, an implicit time stepping scheme is then invoked. Simulations performed for different benchmark test problems feature good agreement with exact solutions. Stability analysis of the proposed method is theoretically discussed and computationally validated to support the analysis. Accuracy and efficiency of the proposed method are assessed via [Formula: see text], [Formula: see text] and [Formula: see text] error norms as well as number of nodes [Formula: see text] and time step-size [Formula: see text].

scholarly journals Efficient extrapolation methods for electro- and magnetoquasistatic field simulations

2003 ◽  
Vol 1 ◽  
pp. 81-86 ◽  
Author(s):  
M. Clemens ◽  
M. Wilke ◽  
T. Weiland
Keyword(s):  
Taylor Series Expansion ◽  
Iterative Solvers ◽  
Gradient Algorithm ◽  
Implicit Time ◽  
Preconditioned Conjugate Gradient ◽  
Test Problems ◽  
Time Step ◽  
Time Stepping ◽  
Multi Stage ◽  
Runge Kutta

Abstract. In magneto- and electroquasi-static time domain simulations with implicit time stepping schemes the iterative solvers applied to the large sparse (non-)linear systems of equations are observed to converge faster if more accurate start solutions are available. Different extrapolation techniques for such new time step solutions are compared in combination with the preconditioned conjugate gradient algorithm. Simple extrapolation schemes based on Taylor series expansion are used as well as schemes derived especially for multi-stage implicit Runge-Kutta time stepping methods. With several initial guesses available, a new subspace projection extrapolation technique is proven to produce an optimal initial value vector. Numerical tests show the resulting improvements in terms of computational efficiency for several test problems. In quasistatischen elektromagnetischen Zeitbereichsimulationen mit impliziten Zeitschrittverfahren zeigt sich, dass die iterativen Lösungsverfahren für die großen dünnbesetzten (nicht-)linearen Gleichungssysteme schneller konvergieren, wenn genauere Startlösungen vorgegeben werden. Verschiedene Extrapolationstechniken werden für jeweils neue Zeitschrittlösungen in Verbindung mit dem präkonditionierten Konjugierte Gradientenverfahren vorgestellt. Einfache Extrapolationsverfahren basierend auf Taylorreihenentwicklungen werden ebenso benutzt wie speziell für mehrstufige implizite Runge-Kutta-Verfahren entwickelte Verfahren. Sind verschiedene Startlösungen verfügbar, so erlaubt ein neues Unterraum-Projektion- Extrapolationsverfahren die Konstruktion eines optimalen neuen Startvektors. Numerische Tests zeigen die aus diesen Verfahren resultierenden Verbesserungen der numerischen Effizienz.

scholarly journals Image Edge Detector with Gabor Type Filters Using a Spiking Neural Network of Biologically Inspired Neurons

Algorithms ◽  
2020 ◽  
Vol 13 (7) ◽  
pp. 165 ◽  
Author(s):  
Krishnamurthy V. Vemuru
Keyword(s):  
Neural Network ◽  
Edge Detection ◽  
Image Classification ◽  
Spiking Neural Network ◽  
Edge Detector ◽  
Time Step ◽  
Biologically Inspired ◽  
Total Exposure Time ◽  
Canny Edge ◽  
Low Exposure

We report the design of a Spiking Neural Network (SNN) edge detector with biologically inspired neurons that has a conceptual similarity with both Hodgkin-Huxley (HH) model neurons and Leaky Integrate-and-Fire (LIF) neurons. The computation of the membrane potential, which is used to determine the occurrence or absence of spike events, at each time step, is carried out by using the analytical solution to a simplified version of the HH neuron model. We find that the SNN based edge detector detects more edge pixels in images than those obtained by a Sobel edge detector. We designed a pipeline for image classification with a low-exposure frame simulation layer, SNN edge detection layers as pre-processing layers and a Convolutional Neural Network (CNN) as a classification module. We tested this pipeline for the task of classification with the Digits dataset, which is available in MATLAB. We find that the SNN based edge detection layer increases the image classification accuracy at lower exposure times, that is, for 1 < t < T /4, where t is the number of milliseconds in a simulated exposure frame and T is the total exposure time, with reference to a Sobel edge or Canny edge detection layer in the pipeline. These results pave the way for developing novel cognitive neuromorphic computing architectures for millisecond timescale detection and object classification applications using event or spike cameras.

A Convex Cutting Plane Algorithm for Global Solution of Generalized Polynomial Optimal Design Models

1996 ◽  
Vol 118 (1) ◽  
pp. 82-88 ◽  
Author(s):  
Chihsiung Lo ◽  
P. Y. Papalambros
Keyword(s):  
Global Optimization ◽  
Cutting Plane ◽  
Test Problems ◽  
Cutting Plane Algorithm ◽  
Computational Results ◽  
Design Models ◽  
Generalized Polynomial ◽  
Speed Reducer ◽  
Improved Performance ◽  
Search Approach

Global optimization algorithms for generalized polynomial design models using a global feasible search approach was discussed in a previous article. A new convex cutting plane algorithm (CONCUT) based on global feasible search and with improved performance is presented in this sequel article. Computational results of the CONCUT algorithm compared to one using linear cuts (LINCUT) are given for various test problems. A speed reducer design example illustrates the application of the algorithms.

Well-balanced compressible cut-cell simulation of atmospheric flow

2009 ◽  
Vol 367 (1907) ◽  
pp. 4559-4575 ◽  
Author(s):  
R. Klein ◽  
K. R. Bates ◽  
N. Nikiforakis
Keyword(s):  
Attractive Alternative ◽  
Test Problems ◽  
Time Step ◽  
Atmospheric Flow ◽  
Flow Equations ◽  
Cut Cell ◽  
Flux Approximation ◽  
Terrain Following ◽  
Lee Wave ◽  
Cell Simulation

Cut-cell meshes present an attractive alternative to terrain-following coordinates for the representation of topography within atmospheric flow simulations, particularly in regions of steep topographic gradients. In this paper, we present an explicit two-dimensional method for the numerical solution on such meshes of atmospheric flow equations including gravitational sources. This method is fully conservative and allows for time steps determined by the regular grid spacing, avoiding potential stability issues due to arbitrarily small boundary cells. We believe that the scheme is unique in that it is developed within a dimensionally split framework, in which each coordinate direction in the flow is solved independently at each time step. Other notable features of the scheme are: (i) its conceptual and practical simplicity, (ii) its flexibility with regard to the one-dimensional flux approximation scheme employed, and (iii) the well-balancing of the gravitational sources allowing for stable simulation of near-hydrostatic flows. The presented method is applied to a selection of test problems including buoyant bubble rise interacting with geometry and lee-wave generation due to topography.

scholarly journals An online trajectory module (version 1.0) for the nonhydrostatic numerical weather prediction model COSMO

2013 ◽  
Vol 6 (6) ◽  
pp. 1989-2004 ◽  
Author(s):  
A. K. Miltenberger ◽  
S. Pfahl ◽  
H. Wernli
Keyword(s):  
High Resolution ◽  
Wind Field ◽  
Weather Prediction ◽  
Limited Area ◽  
Model Output ◽  
Wind Fields ◽  
Time Step ◽  
Flow Distortion ◽  
Computational Costs ◽  
Cosmo Model

Abstract. A module to calculate online trajectories has been implemented into the nonhydrostatic limited-area weather prediction and climate model COSMO. Whereas offline trajectories are calculated with wind fields from model output, which is typically available every one to six hours, online trajectories use the simulated resolved wind field at every model time step (typically less than a minute) to solve the trajectory equation. As a consequence, online trajectories much better capture the short-term temporal fluctuations of the wind field, which is particularly important for mesoscale flows near topography and convective clouds, and they do not suffer from temporal interpolation errors between model output times. The numerical implementation of online trajectories in the COSMO-model is based upon an established offline trajectory tool and takes full account of the horizontal domain decomposition that is used for parallelization of the COSMO-model. Although a perfect workload balance cannot be achieved for the trajectory module (due to the fact that trajectory positions are not necessarily equally distributed over the model domain), the additional computational costs are found to be fairly small for the high-resolution simulations described in this paper. The computational costs may, however, vary strongly depending on the number of trajectories and trace variables. Various options have been implemented to initialize online trajectories at different locations and times during the model simulation. As a first application of the new COSMO-model module, an Alpine north foehn event in summer 1987 has been simulated with horizontal resolutions of 2.2, 7 and 14 km. It is shown that low-tropospheric trajectories calculated offline with one- to six-hourly wind fields can significantly deviate from trajectories calculated online. Deviations increase with decreasing model grid spacing and are particularly large in regions of deep convection and strong orographic flow distortion. On average, for this particular case study, horizontal and vertical positions between online and offline trajectories differed by 50–190 km and 150–750 m, respectively, after 24 h. This first application illustrates the potential for Lagrangian studies of mesoscale flows in high-resolution convection-resolving simulations using online trajectories.

Complementarity Techniques for Minimal Coordinate Contact Dynamics

2016 ◽  
Vol 12 (2) ◽  
Author(s):  
Harshavardhan Mylapilli ◽  
Abhinandan Jain
Keyword(s):  
Complementarity Problem ◽  
Optimization Problems ◽  
Nonlinear Complementarity Problem ◽  
Computational Cost ◽  
Contact Dynamics ◽  
Time Step ◽  
Frictional Contacts ◽  
Computational Costs ◽  
Simulation Results ◽  
Dynamics Problems

In this paper, nonsmooth contact dynamics of articulated rigid multibody systems is formulated as a complementarity problem. Minimal coordinate (MC) formulation is used to derive the dynamic equations of motion as it provides significant computational cost benefits and leads to a smaller-sized complementarity problem when compared with the frequently used redundant coordinate (RC) formulation. Additionally, an operational space (OS) formulation is employed to take advantage of the low-order structure-based recursive algorithms that do not require mass matrix inversion, leading to a further reduction in these computational costs. Based on the accuracy with which Coulomb's friction cone is modeled, the complementarity problem can be posed either as a linear complementarity problem (LCP), where the friction cone is approximated using a polygon, or as a nonlinear complementarity problem (NCP), where the friction cone is modeled exactly. Both formulations are studied in this paper. These complementarity problems are further recast as nonsmooth unconstrained optimization problems, which are solved by employing a class of Levenberg–Marquardt (LM) algorithms. The necessary theory detailing these techniques is discussed and five solvers are implemented to solve contact dynamics problems. A simple test case of a sphere moving on a plane surface is used to validate these solvers for a single contact, whereas a 12-link complex pendulum example is chosen to compare the accuracy of the solvers for the case of multiple simultaneous contacts. The simulation results validate the MC-based NCP formulations developed in this paper. Moreover, we observe that the LCP solvers deliver accuracy comparable to that of the NCP solvers when the friction cone direction vectors in the contact tangent plane are aligned with the sliding contact velocity at each time step. The theory and simulation results show that the NCP approach can be seamlessly recast into an MC OS formulation, thus allowing for accurate modeling of frictional contacts, while at the same time reducing overall computational costs associated with contact and collision dynamics problems in articulated rigid body systems.

scholarly journals Numerical solutions of higher order boundary value problems via wavelet approach

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shams Ul Arifeen ◽  
Sirajul Haq ◽  
Abdul Ghafoor ◽  
Asad Ullah ◽  
Poom Kumam ◽  
...  
Keyword(s):  
Boundary Value Problems ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Linear Equations ◽  
Haar Wavelet ◽  
Higher Order ◽  
Test Problems ◽  
Nonlinear Boundary Value Problems ◽  
Resolution Level ◽  
Linear And Nonlinear

AbstractThis paper presents a numerical scheme based on Haar wavelet for the solutions of higher order linear and nonlinear boundary value problems. In nonlinear cases, quasilinearization has been applied to deal with nonlinearity. Then, through collocation approach computing solutions of boundary value problems reduces to solve a system of linear equations which are computationally easy. The performance of the proposed technique is portrayed on some linear and nonlinear test problems including tenth, twelfth, and thirteen orders. Further convergence of the proposed method is investigated via asymptotic expansion. Moreover, computed results have been matched with the existing results, which shows that our results are comparably better. It is observed from convergence theoretically and verified computationally that by increasing the resolution level the accuracy also increases.

Entropy stabilization and property-preserving limiters for ℙ1 discontinuous Galerkin discretizations of scalar hyperbolic problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Dmitri Kuzmin
Keyword(s):  
Discontinuous Galerkin ◽  
Piecewise Linear ◽  
Maximum Principles ◽  
Test Problems ◽  
Hyperbolic Problems ◽  
Time Step ◽  
Local Maxima ◽  
Slope Limiter ◽  
Entropy Stable ◽  
Flux Limiters

AbstractThe methodology proposed in this paper bridges the gap between entropy stable and positivity-preserving discontinuous Galerkin (DG) methods for nonlinear hyperbolic problems. The entropy stability property and, optionally, preservation of local bounds for cell averages are enforced using flux limiters based on entropy conditions and discrete maximum principles, respectively. Entropy production by the (limited) gradients of the piecewise-linear DG approximation is constrained using Rusanov-type entropy viscosity. The Taylor basis representation of the entropy stabilization term reveals that it penalizes the solution gradients in a manner similar to slope limiting and requires implicit treatment to avoid severe time step restrictions. The optional application of a vertex-based slope limiter constrains the DG solution to be bounded by local maxima and minima of the cell averages. Numerical studies are performed for two scalar two-dimensional test problems with nonlinear and nonconvex flux functions.

Sign in / Sign up

Export Citation Format

Share Document